Biomedical Engineering Reference
In-Depth Information
a
b
c
1
V
= 20
m
eV
V=0
D
=2.68
m
eV,V= 2.97μeV
D
=4
m
eV
0.8
Δ=0
V
= 0.8
m
eV
D
=0.8
μ
eV
exp[-
G
t]
exp[-2
G
t]
0.6
eV
D
=20 μeV
D
=2
μ
Δ = 2 μeV
0.4
V
= 2
m
eV
0.2
V
= 0.8
m
eV
0
0
1
2
3
4
0
1
2
3
4
5
6
0
0.5
1
1.5
2
t
[ns]
t
[ns]
t
[ns]
Fig. 9.11
(
a
)and(
b
) The fidelity for a subradiant initial state for a system of uncoupled QDs
for a few values of the energy mismatch (
a
) and for a coupled system with the energy mismatch
Δ
=
2
μ
eV. (
c
) The fidelity for a superradiant initial state for a constant energy splitting
E
=
8
μ
eV
compared to the exponential decay with decay rates
Γ
and 2
Γ
[
61
]
9.4.1
Stability of Sub- and Superradiant States
The dynamics of a quantum system is governed by the Schr odinger equation which,
for a superposition state, leads to a system of equations for amplitudes of the
contributing states. The Hamiltonian in the rotating wave approximation conserves
the number of excitations (excitons plus photons), thus the evolution of a system
initially prepared in a sub- or superradiant state is governed by a system of three
equations for single exciton amplitudes of the localized states
|
10
and
|
01
and an
amplitude of the state
. This system of equations may be solved using the
Weisskopf-Wigner method [
118
] which is equivalent to description with the Master
equation in the Lindblad form discussed in Sect.
9.2.2.4
. For more details, see [
119
]
and [
61
].
Using the Weisskopf-Wigner method we will solve the evolution equations for
a DQD system and test the stability of sub- and superradiant states spanned in
an
inhomog
eneous DQD. In order to do this we will analyze the fidelity
F
|
00
,
λ
k
=
Φ
|
ρ
|
Φ
between the actual state
ρ
(
t
)
of the exciton subsystem and the pure
state
|
Φ
(
t
)
evolving from the initial state in the absence of the electromagnetic
field
.InFig.
9.11
a we show the evolution of the fidelity for DQD system
initially prepared in the subradiant state
(
Γ
=
0
)
in the limit of vanishing coupling
between the dots (sufficiently distant dots) [
61
]. For atomic-like systems (
|−
Δ
=
0)
the fidelity is stable, since the state
is the eigenstate of the exciton Hamiltonian
of identical dots. This property is extremely sensitive to the homogeneity of the
transition energy and is destroyed for energy splittings of the order of transition line
width (
h
|−
eV). As can be seen, the energy mismatch induces transitions
between sub- and superradiant regimes which lead to oscillations around the
exponential (uncorrelated) decay. The state
Γ
=
0
.
658
μ
|−
maintains its stable character until
t
, and then the system enters the superradiant regimes, i.e., the fidelity
rapidly drops below the exponential decay to return to the subradiant phase after
time
t
∼
π
h
/
(
2
Δ
)
∼
π
h
/
Δ
. While the energy mismatch increases the number of oscillations also
